The precision of that statement can be quibbled, reasonably, IMO. A shutdown that prevented hospitals from being overwhelmed would mean the patients the hospitals did receive would be cared for better, with fewer deaths than if the hospitals had been allowed to be overwhelmed. And that was the original reason given for the shutdowns. But then the goalposts got moved. Another quibble of the precision of the claim is that much was learned about proper treatment, so that delaying people getting infected would mean that when they did, hospitals would know better how to treat them, again, saving some lives.
On the other hand, if a somewhat broader brush is used, long-term shutdowns basically delay the process of herd immunity development, in particular the % that would be developed before a vaccine was approved and became available. So the statement is imprecise, IMO, but still largely true.